Phy331 l3

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PHY331 Magnetism Lecture 3


Last week… •  Derived magnetic dipole moment of a circulating electron. •  Discussed motion of a magnetic dipole in a constant magnetic field. •  Showed that it precesses with a frequency called the Larmor precessional frequency


This week… •  Discuss Langevin’s theory of diamagnetism. •  Use angular momentum of precessing electron in magnetic field to derive the magnetization of a sample and thus diamagnetic susceptibility. •  Will set the scene for the calculation of paramagnetic susceptibility.


Langevin’s theory of diamagnetism •  We want to calculate the sample magnetisation M and the diamagnetic susceptibility χ (recall M = χH) •  Every circulating electron on every atom has a magnetic dipole moment •  The sum of the magnetic dipole moments on any atom is zero (equal numbers circulating clockwise and anticlockwise ?) •  The magnetisation M arises from the reaction to the torque Γ due to the applied magnetic field B which creates the Lamor precessional motion at frequency ωL.


•  i) the angular momentum L of the circulating electron is, 2

L =

mvr =

mω r

•  The angular momentum Lp of the precessional motion is, L p = mω L r 2

where ωL is the Larmor frequency and <r2> is the mean square distance from an axis through the nucleus which is parallel to B


•  ii) We showed last week that the magnetic moment mp per electron associated with Lp is

e mp = − Lp 2m •  so the total magnetisation M of the sample must be, € M = Natoms × Zelectrons/atom × mp M = N × Z × mp


So substitute for each quantity, in turn,

e M =− NZ Lp 2m

e M =− NZ mω L r 2 2m ⎛ eB ⎞ 2 e M =−NZ m ⎜ ⎟ r 2m ⎝ 2m ⎠

so that,

χ =

€ €

M H

2

e = − µ0N Z r2 4m


Now express <r2> in terms of the mean square radius of the orbit <ρ2> , where

r

2

2 2 = ρ 3

will give

2

e χ = − µ0N Z ρ2 € 6m € €

€ The result we wanted!!

2

2

ρ =x +y +z x 2 = y 2 = z 2 = ρ 2 /3 2

2

2

2

r =x +y

2


What does this mean??? •  The atom as a whole does not possess a permanent magnetic dipole moment. •  The magnetisation is produced by the influence of the external magnetic field on the electron orbits and particularly by the precessional motion. •  The diamagnetic susceptibility χ is small and negative, because <ρ2> is small. •  Negative susceptability means that diamagnetism opposes applied magnetic field. (prefix: dia - in opposite or different directions) •  Note that measurements of χ were once used to get estimates of atomic size through the values of <ρ2>.

Remember - all materials are diamagnetic – but this small effect is swamped if also paramagnetic of ferromagnetic.


Paramagnetism •  Paramagnets have a small positive magnetisation M (directed parallel the applied field B). •  Each atom has a permanent magnetic dipole moment. •  Langevin (classical) theory •  The paramagnet consists of an array of permanent magnetic dipoles m •  In a uniform field B they have Potential Energy = - m .B


A dipole parallel to the field has the lowest energy •  BUT, the B field causes precession of m about B. •  However it can’t alter the angle between m and B (as the Lz component is constant in the precession equations). •  For the dipole to lower its energy (and become parallel to the field) we need a second mechanism. This is provided by the thermal vibrations •  The magnetic field “would like” the dipoles aligned to lower their energy •  The thermal vibrations “would like” to randomise and disorder the magnetic dipoles


•  We can therefore expect a statistical theory, based on a “competition” between these two mechanisms,

k BT ≈ −m.B to calculate the magnetisation M of a paramagnet


Calculate the magnetisation M of a paramagnet •  M must be the vector sum of all the magnetic dipole moments

M =∑

[1] Resolved component of a dipole in field direction

X

[2] Number of dipoles with this orientation

To do this, use Boltzmann statistics to obtain the number dn of dipoles with energy between E and E + dE

dn = c exp (−E kT ) dE

where k = Boltzmann’s constant, c = is a constant of the system and T = T measured in Kelvin


How to find c? Integrate over all the energies, which must give N the total number of dipoles ∞

N=

∫ c exp (−E

kT ) dE

0 Next week we will do this to calculate the susceptibility of a paramagnet. The important result we will get will be That susceptibility is temperature dependent with

χ = C/T

This is Curie’s law.


Summary •  Calculated the susceptibility for a diamagnet e2 2 χ = − µ0N Z ρ

6m

•  Argued that all materials have a small diamagnetic susceptibility resulting from the precession of electrons in a magnetic field. € precession results in a magnetic moment, and •  This thus a susceptibility. •  Then discussed paramagnetism, resulting from atoms having a permanent magnetic moment. •  Sketched out how we will calculate paramagnetic susceptibility.


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